My Math M.Sc. Thesis: Hilbert Schemes and the Derived McKay Correspondence
The Math Thesis
The full title is “The behavior of the Hilbert scheme of points under the derived McKay correspondence” and it’s now in the UBC Library Open Collections.
The Thesis
Can’t see the embed? View the thesis on UBC Library →
What’s It About?
This one is pure algebraic geometry. Let me try to explain the main ideas without losing everyone.
The Setup
Start with \(\mathbb{C}^2\) (the complex plane, two copies) and quotient it by a cyclic group \(\mathbb{Z}/n\mathbb{Z}\). This creates a singularity, a Kleinian quotient singularity to be precise. These singularities have a beautiful structure and are classified by ADE Dynkin diagrams.
Now, you can “resolve” this singularity, meaning you replace the bad point with something smooth. The minimal crepant resolution \(Y\) is particularly nice because it doesn’t introduce any extra “canonical” badness.
The McKay Correspondence
The McKay correspondence is one of those magical results in math that connects seemingly unrelated things:
- Representation theory of the group \(\mathbb{Z}/n\mathbb{Z}\)
- Geometry of the resolution \(Y\)
- Combinatorics of Dynkin diagrams
The derived McKay correspondence takes this further. It says there’s an equivalence of categories:
\[D^b(\text{Coh}(Y)) \cong D^b_{\mathbb{Z}/n}(\text{Coh}(\mathbb{C}^2))\]
The derived category of coherent sheaves on \(Y\) is equivalent to the equivariant derived category on \(\mathbb{C}^2\). This equivalence is given by a Fourier-Mukai transform, a powerful tool in algebraic geometry.
The Main Result
My thesis completely determines what happens to the Hilbert scheme of points under this equivalence.
Specifically, I look at structure sheaves of zero-dimensional, torus-invariant closed subschemes on \(Y\) and track their images under the Fourier-Mukai transform.
The punchline? There’s a beautiful combinatorial correspondence:
Partitions ↔︎ \(\mathbb{Z}/n\)-colored skew partitions
If you know about Young diagrams and partition combinatorics, this is a satisfying result. The geometry translates into pure combinatorics.
Why I Love This Stuff
There’s something deeply satisfying about derived categories. They’re abstract, sure. But they reveal structure that you can’t see otherwise.
The McKay correspondence is a perfect example. On the surface, you have a group acting on a space. Underneath, there’s this rich tapestry connecting algebra, geometry, and combinatorics. The derived perspective makes it all fit together.
The Journey
This was my first thesis. I spent months reading papers, filling notebooks with diagrams, and trying to understand Bridgeland’s work on derived categories. There were weeks where I felt completely lost.
But when it clicked? When I finally saw how the Fourier-Mukai kernel worked, how the exceptional divisors corresponded to representations, how the combinatorics emerged from the geometry? That was worth every confused hour.
Math is like that. Long periods of confusion punctuated by moments of clarity.